the way to Jupiter yielded no anomalous acceleration. In 2018, a careful analysis of the trajectory of the presumed interstellar asteroid ʻOumuamua revealed a small excess velocity as it receded from the Sun. Initial speculation suggested that the anomaly was due to outgassing, though none had been detected. Summary of some Earth-flyby spacecraft is provided in table below. Anderson's empirical relation An empirical equation for the anomalous flyby velocity change was proposed in 2008 by J. D. Anderson et al.: where ωE is the angular frequency of the Earth, RE is the Earth radius, and φi and φo are the inbound and outbound equatorial angles of the spacecraft. This formula was derived later by Jean Paul Mbelek from special relativity, leading to one of the possible explanations of the effect. This does not, however, consider the SSN residuals – see "Possible explanations" below. Possible explanations There have been a number of proposed explanations of the flyby anomaly, including: A postulated consequence of the assumption that the speed of light is isotropic in all frames, and invariant in the method used to measure the velocity of the space probes by means of the Doppler Effect. The inconsistent anomalous values measured: positive, null or negative are simply explained relaxing this assumption. During flyby maneuvers the velocity components of the probe in the direction of the observer V are derived from the relative displacement df of the radiofrequency f transmitted by the probe, multiplied by the local speed of the light c′ by the Doppler effect: V = (df / f) c′. According to the Céspedes-Curé hypothesis, the movement through variable gravitational energy density fields produces slight variations of the refractive index n′ of space and therefore of the speed of light c′ which leads to unaccounted corrections of the Doppler data that are based on an invariant c. This leads to incorrect estimates of the speed or energy change in the flyby maneuver on the Earth's frame of reference. Unaccounted-for transverse Doppler effect—i.e. the redshift of light source with zero radial and non-zero tangential velocity. However, this cannot explain the similar anomaly in the ranging data. A dark-matter halo around Earth. The impact of general relativity, in its weak-field and linearized form yielding gravitomagnetic phenomena like frame-dragging, has been investigated as well: it turns out to be unable to account for the flyby anomaly. The classical time-retarded gravity explanation proposed by Joseph